Question

for the rotation $-249^{circ}$, find the coterminal angle from $0^{circ} \leq \theta < 360^{circ}$, the quadrant, and the reference angle.

the coterminal angle is \\(\square^{\circ}\\), which lies in quadrant \\(\square\\), with a reference angle of \\(\square^{\circ}\\).

Explanation:

Step1: Find coterminal angle

To find a coterminal angle with \(-249^\circ\) in the range \(0^\circ\leq\theta < 360^\circ\), we add \(360^\circ\) (since adding or subtracting multiples of \(360^\circ\) gives coterminal angles).
\(-249^\circ+ 360^\circ=111^\circ\)

Step2: Determine the quadrant

Angles between \(90^\circ\) and \(180^\circ\) lie in the second quadrant. Since \(90^\circ<111^\circ < 180^\circ\), \(111^\circ\) is in the second quadrant.

Step3: Find the reference angle

For an angle \(\theta\) in the second quadrant, the reference angle \(r\) is given by \(r = 180^\circ-\theta\).
Substituting \(\theta = 111^\circ\), we get \(r=180^\circ - 111^\circ = 69^\circ\)

Answer:

The coterminal angle is \(111^\circ\), which lies in Quadrant II, with a reference angle of \(69^\circ\).